Deforming the Maxwell-Sim Algebra

TL;DR

This paper extends the Maxwell algebra within the context of Very Special Relativity, deriving particle actions that describe interactions with electromagnetic fields, including Finslerian modifications, through algebraic deformations.

Contribution

It introduces a Maxwell extension of the ISim algebra and explores its deformation to DISim_b, linking algebraic structures to particle dynamics with electromagnetic interactions.

Findings

Maxwell-Sim algebra describes particles under Lorentz force.

Deformation to Maxwell-DISim yields Finslerian Lorentz force modifications.

Particle actions correspond to specific algebraic symmetries.

Abstract

The Maxwell alegbra is a non-central extension of the Poincar\'e algebra, in which the momentum generators no longer commute, but satisfy $[P_\mu,P_\nu]=Z_{\mu\nu}$. The charges $Z_{\mu\nu}$ commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincar\'e, this being the symmetry algebra of Very Special Relativity. It admits an analogous non-central extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim$_b$, where $b$ is a non-trivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding Maxwell-DISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force.